3.1900 \(\int \frac{(d+e x)^6}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\)

Optimal. Leaf size=35 \[ -\frac{(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3} \]

[Out]

-(d + e*x)^3/(3*(c*d^2 - a*e^2)*(a*e + c*d*x)^3)

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Rubi [A]  time = 0.0125113, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 37} \[ -\frac{(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

-(d + e*x)^3/(3*(c*d^2 - a*e^2)*(a*e + c*d*x)^3)

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=\int \frac{(d+e x)^2}{(a e+c d x)^4} \, dx\\ &=-\frac{(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0315722, size = 65, normalized size = 1.86 \[ -\frac{a^2 e^4+a c d e^2 (d+3 e x)+c^2 d^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 c^3 d^3 (a e+c d x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

-(a^2*e^4 + a*c*d*e^2*(d + 3*e*x) + c^2*d^2*(d^2 + 3*d*e*x + 3*e^2*x^2))/(3*c^3*d^3*(a*e + c*d*x)^3)

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Maple [B]  time = 0.044, size = 96, normalized size = 2.7 \begin{align*} -{\frac{{a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{3\,{c}^{3}{d}^{3} \left ( cdx+ae \right ) ^{3}}}+{\frac{e \left ( a{e}^{2}-c{d}^{2} \right ) }{{c}^{3}{d}^{3} \left ( cdx+ae \right ) ^{2}}}-{\frac{{e}^{2}}{{c}^{3}{d}^{3} \left ( cdx+ae \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)

[Out]

-1/3*(a^2*e^4-2*a*c*d^2*e^2+c^2*d^4)/c^3/d^3/(c*d*x+a*e)^3+e*(a*e^2-c*d^2)/c^3/d^3/(c*d*x+a*e)^2-e^2/c^3/d^3/(
c*d*x+a*e)

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Maxima [B]  time = 1.09496, size = 153, normalized size = 4.37 \begin{align*} -\frac{3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \,{\left (c^{6} d^{6} x^{3} + 3 \, a c^{5} d^{5} e x^{2} + 3 \, a^{2} c^{4} d^{4} e^{2} x + a^{3} c^{3} d^{3} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima")

[Out]

-1/3*(3*c^2*d^2*e^2*x^2 + c^2*d^4 + a*c*d^2*e^2 + a^2*e^4 + 3*(c^2*d^3*e + a*c*d*e^3)*x)/(c^6*d^6*x^3 + 3*a*c^
5*d^5*e*x^2 + 3*a^2*c^4*d^4*e^2*x + a^3*c^3*d^3*e^3)

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Fricas [B]  time = 1.81493, size = 221, normalized size = 6.31 \begin{align*} -\frac{3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \,{\left (c^{6} d^{6} x^{3} + 3 \, a c^{5} d^{5} e x^{2} + 3 \, a^{2} c^{4} d^{4} e^{2} x + a^{3} c^{3} d^{3} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")

[Out]

-1/3*(3*c^2*d^2*e^2*x^2 + c^2*d^4 + a*c*d^2*e^2 + a^2*e^4 + 3*(c^2*d^3*e + a*c*d*e^3)*x)/(c^6*d^6*x^3 + 3*a*c^
5*d^5*e*x^2 + 3*a^2*c^4*d^4*e^2*x + a^3*c^3*d^3*e^3)

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Sympy [B]  time = 1.31524, size = 121, normalized size = 3.46 \begin{align*} - \frac{a^{2} e^{4} + a c d^{2} e^{2} + c^{2} d^{4} + 3 c^{2} d^{2} e^{2} x^{2} + x \left (3 a c d e^{3} + 3 c^{2} d^{3} e\right )}{3 a^{3} c^{3} d^{3} e^{3} + 9 a^{2} c^{4} d^{4} e^{2} x + 9 a c^{5} d^{5} e x^{2} + 3 c^{6} d^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

-(a**2*e**4 + a*c*d**2*e**2 + c**2*d**4 + 3*c**2*d**2*e**2*x**2 + x*(3*a*c*d*e**3 + 3*c**2*d**3*e))/(3*a**3*c*
*3*d**3*e**3 + 9*a**2*c**4*d**4*e**2*x + 9*a*c**5*d**5*e*x**2 + 3*c**6*d**6*x**3)

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Giac [B]  time = 90.6134, size = 1110, normalized size = 31.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")

[Out]

-1/3*(3*c^8*d^14*x^5*e^5 + 12*c^8*d^15*x^4*e^4 + 19*c^8*d^16*x^3*e^3 + 15*c^8*d^17*x^2*e^2 + 6*c^8*d^18*x*e +
c^8*d^19 - 18*a*c^7*d^12*x^5*e^7 - 69*a*c^7*d^13*x^4*e^6 - 104*a*c^7*d^14*x^3*e^5 - 78*a*c^7*d^15*x^2*e^4 - 30
*a*c^7*d^16*x*e^3 - 5*a*c^7*d^17*e^2 + 45*a^2*c^6*d^10*x^5*e^9 + 162*a^2*c^6*d^11*x^4*e^8 + 226*a^2*c^6*d^12*x
^3*e^7 + 156*a^2*c^6*d^13*x^2*e^6 + 57*a^2*c^6*d^14*x*e^5 + 10*a^2*c^6*d^15*e^4 - 60*a^3*c^5*d^8*x^5*e^11 - 19
5*a^3*c^5*d^9*x^4*e^10 - 236*a^3*c^5*d^10*x^3*e^9 - 138*a^3*c^5*d^11*x^2*e^8 - 48*a^3*c^5*d^12*x*e^7 - 11*a^3*
c^5*d^13*e^6 + 45*a^4*c^4*d^6*x^5*e^13 + 120*a^4*c^4*d^7*x^4*e^12 + 100*a^4*c^4*d^8*x^3*e^11 + 30*a^4*c^4*d^9*
x^2*e^10 + 15*a^4*c^4*d^10*x*e^9 + 10*a^4*c^4*d^11*e^8 - 18*a^5*c^3*d^4*x^5*e^15 - 27*a^5*c^3*d^5*x^4*e^14 + 1
6*a^5*c^3*d^6*x^3*e^13 + 30*a^5*c^3*d^7*x^2*e^12 - 6*a^5*c^3*d^8*x*e^11 - 11*a^5*c^3*d^9*e^10 + 3*a^6*c^2*d^2*
x^5*e^17 - 6*a^6*c^2*d^3*x^4*e^16 - 26*a^6*c^2*d^4*x^3*e^15 - 12*a^6*c^2*d^5*x^2*e^14 + 15*a^6*c^2*d^6*x*e^13
+ 10*a^6*c^2*d^7*e^12 + 3*a^7*c*d*x^4*e^18 + 4*a^7*c*d^2*x^3*e^17 - 6*a^7*c*d^3*x^2*e^16 - 12*a^7*c*d^4*x*e^15
- 5*a^7*c*d^5*e^14 + a^8*x^3*e^19 + 3*a^8*d*x^2*e^18 + 3*a^8*d^2*x*e^17 + a^8*d^3*e^16)/((c^9*d^15 - 6*a*c^8*
d^13*e^2 + 15*a^2*c^7*d^11*e^4 - 20*a^3*c^6*d^9*e^6 + 15*a^4*c^5*d^7*e^8 - 6*a^5*c^4*d^5*e^10 + a^6*c^3*d^3*e^
12)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^3)